Contents

Preface page xi

1 Elliptic functions 11.1 Values of elliptic functions 11.2 The functions σ(z|L), ζ(z |L) and ℘(z |L) 31.3 Construction of elliptic functions 71.4 Algebraic and geometric properties of elliptic functions 91.5 Division polynomials 131.6 Weierstrass functions 16

1.6.1 Expansions at zero 181.6.2 p-adic limits 23

1.7 Elliptic resolvents 271.8 q-expansions 321.9 Dedekind’s η function and σ-product formula 351.10 The transformation formula of the Dedekind η

function 38

2 Modular functions 412.1 The modular group 422.2 Congruence subgroups 452.3 Definition of modular forms 482.4 Examples of modular forms and modular functions 50

2.4.1 The functions g2, g3 and ∆ 502.4.2 The functions j, 3

√j, 2

√j − 123, jR, ϕR 50

2.4.3 η-quotients 512.4.4 Weber’s τ function 522.4.5 The natural normalisation of the ℘ function 532.4.6 Klein’s normalisation of the σ function 532.4.7 Transformation of τ (e), p, ϕ 53

vii

www.cambridge.org© in this web service Cambridge University Press

Cambridge University Press978-0-521-76668-5 - Complex MultiplicationReinhard SchertzTable of ContentsMore information

viii Contents

2.5 Modular functions for Γ 542.5.1 Construction of modular functions for Γ 542.5.2 The q-expansion principle 59

2.6 Modular functions for subgroups of Γ 612.6.1 The isomorphisms of CU/CΓ 612.6.2 The extended q-expansion principle 62

2.7 Modular functions for ΓR 632.8 Modular functions for Γ(N) 692.9 The field Q(γ2, γ3) 722.10 Lower powers of η-quotients 74

3 Basic facts from number theory 823.1 Ideal theory of suborders in a quadratic number field 82

3.1.1 Fractional ideals, integral ideals, properideals, regular ideals 82

3.1.2 Ideal groups 863.1.3 Primitive matrices and bases of ideals 943.1.4 Integral ideals that are not regular 98

3.2 Density theorems 1003.3 Class field theory 103

4 Factorisation of singular values 1114.1 Singular values 1114.2 Factorisation of ϕA(α) 1144.3 Factorisation of ϕ(ξ | L) 1184.4 A result of Dorman, Gross and Zagier 121

5 The Reciprocity Law 1225.1 The Reciprocity Law of Weber, Hasse, Söhngen,

Shimura 1225.2 Applications of the Reciprocity Law 128

6 Generation of ring class fields and ray class fields 1386.1 Generation of ring class fields by singular values of j 1386.2 Generation of ray class fields by τ and j 1416.3 The singular values of γ2 and γ3 1446.4 The singular values of Schläfli’s functions 1486.5 Heegner’s solution of the class number one problem 1516.6 Generation of ring class fields by η-quotients 1546.7 Double η-quotients in the ramified case 1656.8 Generation of ray class fields by ϕ(z|ω1

ω2) 169

6.9 Generalised principal ideal theorem 183

www.cambridge.org© in this web service Cambridge University Press

Cambridge University Press978-0-521-76668-5 - Complex MultiplicationReinhard SchertzTable of ContentsMore information

Contents ix

7 Integral basis in ray class fields 1907.1 A normalisation of the Weierstrass ℘ function 1917.2 The discriminant of P(δ) 1937.3 The denominator of P(δ) 1977.4 Construction of relative integral basis 201

7.4.1 Analogy to cyclotomic fields 2037.5 Relative integral power basis 2057.6 Bley’s generalisation for Kt,f/Ωt with t > 1 210

8 Galois module structure 2138.1 Torsion points and good reduction 2148.2 Kummer theory of E 2158.3 Integral objects 2178.4 Global construction of OP and A as OL-algebras 2208.5 Construction of a generating element for OP over A 2218.6 Galois module structure of ray class fields 2248.7 Models of elliptic curves 228

8.7.1 The Weierstrass model 2288.7.2 The Fueter model 2298.7.3 The Deuring model 2318.7.4 Singular values of the Weierstrass, Fueter

and Deuring functions 2328.7.5 Singular values of Weierstrass functions 234

8.8 Proofs of Theorems 8.3.1 and 8.5.1 2388.9 Proofs of Theorems 8.4.1, 8.4.2 and 8.5.2 2458.10 Proofs of Theorems 8.9.2 and 8.6.2 2508.11 Analogy to the cyclotomic case 2538.12 Generalisation to ring classes by Bettner and Bley 256

9 Berwick’s congruences 2619.1 Bettner’s results 2619.2 Method of proof 263

10 Cryptographically relevant elliptic curves 26610.1 Reduction of the Weierstrass model 26610.2 Computation of j(O) modulo P 273

10.2.1 Schläfli–Weber functions 27510.2.2 Double η-quotients 27610.2.3 Application of η-quotients in the ramified case 278

10.3 Reduction of the Fueter and Deuring models 28210.3.1 Reduction of the Fueter model 28210.3.2 Reduction of the Deuring model 285

www.cambridge.org© in this web service Cambridge University Press

Cambridge University Press978-0-521-76668-5 - Complex MultiplicationReinhard SchertzTable of ContentsMore information

x Contents

11 The class number formulae of Curt Meyer 28811.1 L-Functions of ring class characters 28911.2 L-function s of ray class characters χ with fχ = (1). 29111.3 Class number formulae 293

12 Arithmetic interpretation of class number formulae 29512.1 Group-theoretical lemmas for the case L ⊇ K 29512.2 Applications of Theorems 12.1.1, 12.1.2 301

12.2.1 Application of Theorem 12.1.1 30212.2.2 Application of Theorem 12.1.2 303

12.3 Class number formulae for Ω ⊇ L ⊇ K 30412.4 Class number formulae for Kf ⊇ L ⊇ K 309

12.4.1 Application of the formulae from 12.4 31712.5 Group-theoretical lemmas for M ⊇ K 32312.6 The Galois group of MK/K 33612.7 Class number formulae for Ω ⊃ M K 33812.8 Class number formulae for Kf ⊃ M K 341

12.8.1 Applications of the class number formulaein 12.8 346

References 351Index of Notation 356Index 360

www.cambridge.org© in this web service Cambridge University Press

Cambridge University Press978-0-521-76668-5 - Complex MultiplicationReinhard SchertzTable of ContentsMore information